Mathematics plays an important role in two distinct spheres: that of education and scholarship, and that of administration and trade. How much overlap of the ideas, content, techniques, and personnel is there between the worlds of mathematical scholars and those who use mathematics as part of their daily practice? Historians of mathematics tend to focus on mathematical theory more than practice, while economic historians study patterns of trade and administrative structure with less interest in mathematical concepts. The volume under review seeks to probe the questions of overlap and disjunction in the theory and practice of mathematics.

The ancient worlds of the title are primarily the civilizations of Mesopotamia, China, and India, although there is one final contribution on medieval Europe. The volume does not attempt an encyclopedic synthesis. While confronting similar questions, the individual chapters function largely as independent case studies, and the reader may pick and choose items of interest. A useful series of appendices at the end of the volume cover the metrological systems and representation of numbers from Mesopotamia, China, and South Asia.

In the 50-page introduction, the editors lay out their stall. While they see the current volume as very much only a preliminary step on a long journey, the goal is to ``provide a better understanding of the role mathematical knowledge and practices played in allowing various types of practitioners to carry out managerial and economic activities in the ancient worlds.’’ (1) That is, mathematics is an enabling technology for practitioners, but what exactly that mathematics is in any culture can only be revealed by detailed study of the sources. The introduction lays out the theoretical basis of the project and provides a comparison of the types and range of sources available for the study of the different cultures. These vary widely and impose limits on the questions that can be asked in each subsequent chapter. The editors give brief introductions to each chapter, and these introductions are naturally more comparative than the individual investigations. The summative nature of the introduction means that it can be usefully read as a conclusion once the detailed arguments of the following chapters are digested.

The body of the work is divided into four parts. Part I, Mathematical Writings, Regulations, Laws and Norms, contains three chapters. In Chapter 2, Cécile Michel compares prices and wages in royal inscriptions, administrative and mathematical texts from Old Babylonian (ca. 2000-1600 BCE) Larsa. Kings liked to boast of their construction projects and how well the realm was doing under their reign in royal inscriptions. Some of these inscriptions detail wages paid to construction workers and prices of goods. A comparison with actual administrative documents from the same period show that the claimed wages are high and prices are low in the royal inscriptions. Wages and prices in mathematical texts are close to reality, bearing in mind rounding and the exigencies of particular problem calculations.

In the second chapter of Part I, Mark McClish studies computation in the Arthaśāstra. The Arthaśāstra of Kauṭilya (ca. 1st century BCE – 3rd century CE) is one of the most important sources of state administration for the period. Considered a manual of statecraft, it has been studied from many different directions. In this case, McClish approaches the document from the angle of mathematics, considering the metrology it details as well as the content, terminology, and conceptions of arithmetic in the context of state activities. Moving further east to China and widening the time-frame slightly from -300 to +300, Hao Peng considers official salaries and state taxes in Qin-Han manuscripts. As in Larsa, the data from the mathematical texts are similar to actual administrative values. Official salaries were given in terms of grain. However, in a country as large as China, many types of grain are grown, and the value and volume of grain depend upon the stage of processing. Hence, the bureaucracy developed an extensive collection of conversion rates. The computations involved provide fodder for the historian. Similarly, taxes, including taxes on land and customs taxes, could be paid in goods or cash, again requiring various conversions and computations.

Part II, Quantifying Spatial Extension, Quantifying Work, also contains three chapters, two on Mesopotamian sources, one on China. In Chapter 6, on the administration of irrigation systems in Ur III (2112—2004 BCE) Mesopotamia, Stephanie Rost argues that while the detailed bookkeeping procedures of the era did act to provide close oversight of accounting tasks, they were also designed to streamline workflow for the managers on the ground. Rost’s argument rests on reconstructions of ancient irrigation control systems, especially flow-dividers, and an analysis of mid-level managers drawing up accounts.

Turning from waterways to brick construction and spanning Ur III and the subsequent Old Babylonian periods, Martin Sauvage considers the mathematical computations required for the management of public construction works. The focus here is on bricks and labor norms associated with production, movement and construction involving bricks and conversions between different kinds of magnitudes, such as weight and volume. There are two relevant classes of texts, administrative ones, and mathematical ones from a scholarly or scholastic setting. Sun-dried bricks made from clay or clay mixed with straw were a staple of Mesopotamian construction and came in many (standardized) shapes and sizes. Sauvage determines that the cited administrative work-norms for unskilled laborers were realistic, if strenuous, and that the values used in the mathematical texts reflect those in the administrative texts, although given the span of time involved, it is not always clear that mathematical texts reflected contemporary work norms, and not just ones embedded in tradition.

In Chapter 7, Karine Chemla and Biao Ma present an interesting argument for the artificiality of units of volume in measuring grain in Early Imperial China (ca. 300 BCE – 100 CE). Grain is naturally measured in capacity units, and there was a standard decimal system of capacity units in use, although the names of some of the units changed over the period under consideration. In contrast, there were no standard units of volume. Instead, volume was given in length units (representing height) for a rectangular parallelepiped with a standard square base (usually 1 square chi). That is, volumes were not measured, only computed, and were used to determine the value of grain. The sources the authors draw on show a close relationship between the mathematical documents and administrative texts on the management of grain, perhaps not surprising in view of the mathematics underlying the use of volume and value.

Part III, Quantifying Land and Surfaces, turns from grain to the fields that produced the grain and other foodstuffs. Camille Lecompte considers the measurement of fields in Mesopotamia in the 24th century BCE, with most sources coming from pre-Sargonic Girsu, but some also from Umma. Agriculture in Southern Mesopotamia depended on irrigation, and one expects fields to be predominantly long and narrow, with the narrow end against the river or canal, water-front property being expensive. Surveys of fields from the later third millennium show that they largely follow this pattern. In contrast, Lecompte shows that pre-Sargonic agriculture operated with more diverse field shapes, and that this in turn presented surveyors with more complicated calculations. Fields are measured linearly, but the value of interest is the area. Determination of the area of an irregular field from measurement of sides was complicated and Lecompte explores the various techniques of approximations and rounding used by the surveyors. Lecompte also includes useful appendices detailing the corpus of tablets.

Determination of areas of square or rectangular fields is a problem that goes back to the earliest written sources in Mesopotamia. Among the sources are tables linking lengths and areas. There has been considerable debate about the usage of these tables as well as how the underlying calculations were conceived and carried out before the Old Babylonian period. In her chapter, Christine Proust considers a collection of five such tables spanning roughly 2500 to 2350 BCE. She makes a number of astute observations. First, the methods of calculation vary according to the size of the areas being determined, as well as the period in which the computations were made. Proust favors a geometrical approach requiring only simple addition for large areas in the older tablets, and a complex cut and paste geometry for small area computations in the later texts. Additionally, the equivalence between fractions and sexagesimal subdivisions of a basic area unit hint at a proto-reciprocal table. In all, the changing procedures indicating emerging mathematical concepts, especially around multiplication.

Part IV covers Prices, Rates, Loans, and Interests. Using a variety of methodologies, the authors approach these topics from different directions across ancient cultures. Opening the section, Cécile Michel considers the arithmetical techniques of Assyrian merchants. The sexagesimalization of Southern Mesopotamia and the famous sexagesimal (base-60) place-value system of computation with its attendant conversion, multiplication and reciprocal tables had less impact on the north, which had an inherited decimal system. In the 19th century BCE Assyrian merchants were involved in long-distance trade between their home bases in Assyria and Kaneš (modern Kültepe) in Central Anatolia. Between communications with Assyria and daily commercial transactions, they generated thousands of tablets. One standard practice was the conversion of weights of gold, tin, and copper into silver equivalences to compare and record value. Michel shows that the merchants preferred to deal with fractions rather than multiples of smaller metrological units, used subtractive notation, and often dealt with approximations.

Turning to the south, Robert Midekke-Conlin zeroes in on a single Old Babylonian economic document, YBC 7473, probably from Larsa. There is a key error in the conversion of a quantity of sesame into its equivalent in silver. From a detailed analysis, Middeke-Conlin shows that the discrepancy is due to a particular form of rounding that in turn suggests the merchant preparing the document was familiar with the sexagesimal place-value system and the conversion tables from metrological units into sexagesimal equivalents.

In Chapter 12, Sreeramula Rajeswara Sarma and Takamori Kusuba consider the depiction of interest rates on loans in Sanskrit legal and mathematical texts. Interest rates were based on caste as well as the purpose of the loan and there is a question of how closely real interest rate practices are reflected in these texts, especially since the merchants who lent the money had access to neither category of text. It seems that, at least in some cases, the legal texts do give accurate portrayals of interest rates, but that the mathematical texts tend to use hypothetical interest rates and are more interested in developing a wide range of problems, extending simple interest rate problems that were solved using the Rule of Three or Rule of Five.

Rounding out the volume, Marc Bompaire and Matthieu Husson compare John of Murs’ *Quadripartitum Numerorum* with extant 14th and 15th-century French money changers’ books on the subject of coins and coinage. The main issue dealt with is the conversion of a collection of mixed coins into an equivalent value of silver as we have seen in the previous chapters, but also how to generate money of a given target value by changing the proportion of silver or copper in it. In John of Murs’ text, monetary metrology is given by weight. In contrast, in the money changers’ world, coinage is defined by alloy, the proportion of silver it contains. Confusingly, both the unit of weight and the unit of alloy were called deniers. The arithmetical techniques in both sources overlap to an extent, but in the practical works they are designed to facilitate easy mental computation. The authors draw out key distinctions between the theoretical and practical approaches to dealing with money and coinage.

All in all, the chapters in this volume provide a diverse set of insights into the ways that mathematics was used in ancient societies, how practical training was achieved, and the varying distances between mathematical practice and theoretical scholarship. Anyone interested in the social uses of mathematics can read this work for pleasure and profit.

Duncan J. Melville is the Martha E. ’62 and Gregg E. Peterson Professor of Mathematics at St. Lawrence University. He is a historian of mathematics with a specialty in Mesopotamian mathematics.